Is '=' antisymmetric?

Question

I know that an antisymmetric relation must meet the following condition: If x <=y and y<=x then x=y. That being said, can one consider x=x to be antisymmetric?

P.S.: Something tells me that the answer is right under my eyes, but I just cannot seem to wrap my head around it.

Answer 1

A relation $R$ is antisymmetric if the following statement is true for all $x$ and $y$:

If $xRy$ and $yRx$, then $x=y$.

To test whether this is true when $R$ is $=$, you just plug in $=$ for $R$. So the question is, is the following true for all $x$ and $y$?

If $x=y$ and $y=x$, then $x=y$.

Do you now see how to answer the question?

Answer 2

Yes, yes it is.

The definition is as you put it.

Or in other words, an antisymmetic relation is one where there is no distinct pair a, b ( a$\ne$ b) where aRb and bRa. As there are no distinct pair a $\ne$ b where a = b and b = a, equality is by definition antisymmetric.